Ikeda¶

Systems / Maps / Chaotic maps

The Ikeda map — a spiral strange attractor from a nonlinear optical ring cavity.

discrete · map2 dimensionschaotic

Definition¶

@staticmethod
def _step(X, a, b, u):
    x, y = X
    t = a - b / (1 + x**2 + y**2)
    xp = 1 + u * (x * np.cos(t) - y * np.sin(t))
    yp = u * (x * np.sin(t) + y * np.cos(t))
    return xp, yp

Parameters¶

Symbol	Default	Role
`a`	`0.4`	phase-offset constant of the cavity detuning
`b`	`6`	nonlinear phase-modulation strength
`u`	`0.9`	round-trip field-loss (dissipation) factor

State variables: x, y

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

3 fixed points
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Ikeda()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Ikeda (1979), Opt. Commun. 30, 257-261
doi:10.1016/0030-4018(79)90090-7

BibTeX

@misc{ikeda,
  title = {Ikeda system},
  note = {Ikeda (1979), Opt. Commun. 30, 257-261},
  doi = {10.1016/0030-4018(79)90090-7}
}